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Right, three things.
She's not weapons grade. Yet. You haven't met her. You don't know if she picks her nose and eats it, laughs like a donkey, or smells slightly of whisky and sick.
Secondly (and sorry to disagree with previous posters) - don't bring up the fact that she doesn't message you first. She hasn't met you yet. She's not as invested as you are, although she may actually be very keen - but saying that would, I think, make you look needy or politely sulky.
Thirdly - yes, everyone is different. If she wants to be chased, and you're okay to chase, then do it. Eventually she needs to want to chase you back, but you haven't proved yourself yet.
Sort the date out, meet her, and then take the next step when there's a next step to take.
In the meantime, be funny and charming but not obsequious, and make her want to meet you.
Getting the first date is (relatively easy. It's getting the second and third dates that are tricky, because you're suddenly faced with a whole different person from who you thought, and it's no longer a stranger on a screen. Her voice may be nasal. She may get angry and bitter about her ex.
(These two are the reasons I've called things off with last weeks date)
Good luck, padawan:)